Abstracts
Plenary speakers
Randy LeVeque
Title: Coupled Seismic/Hydroacoustic/Tsunami Simulation
Abstract: A widespread approach for tsunami simulation is to
model the ocean with the shallow water equations (SWE). The steady-state
ocean floor deformation that results from a seismic event is often used
as the initial ocean surface profile for the SWE, taking advantage of the
separation of the tsunami and seismic timescales when the event occurs far
from the inundation area of interest. This work focuses instead on tsunamis
generated by near-field seismic events and the data that might be measured
by proposed seafloor sensors for use in tsunami early warning, as part of
a project to study the feasibility and utility of an offshore network of
sensors on the Cascadia Subduction Zone (CSZ). The dynamic seismic/acoustic
and tsunami waves are simulated by modeling the ground as an elastic solid
that is coupled with an acoustic ocean layer. A gravity term is added to
the ocean layer in order to capture the tsunami (gravity waves) that are
generated by the acoustic waves. The AMRClaw and Geoclaw (
www.clawpack.org)
packages are used to simulate a two-dimensional vertical cross section,
resembling CSZ topography, in order to explore this approach. This is
joint work with Chris Vogl at Lawrence Livermore National Laboratory.
Speakers for the Modeling Natural Hazards Session
David George
Title: Fifteen Years of Modeling Shallow Earth-Surface Flows:
the evolution of D-Claw and beyond
Abstract: D-Claw is software developed for modeling landslides
and debris flows, from initiation (accounting for slope-stability of
a perturbed stable sediment mass), to flow mobilization and dynamics,
and finally to deposition. It is based on a two-phase (granular-fluid),
depth-averaged set of PDEs for depth, momentum, volume fractions, and
pore-fluid pressure for flow over general topography. The equations are
hyperbolic balance laws with source terms arising from topography, and are
a more complex analog of the shallow water equations, for which the GeoClaw
software was developed. These software packages are extensions or subsets
of the open-source Clawpack software, originally developed in the Amath
department at UW.
This talk will begin with a brief description of the evolution of D-Claw from
the Clawpack software. The D-Claw model, and the experiments that informed
and motivated its development will be described. Finally, most of the talk
will showcase D-Claw applications related to volcano hazards in the Pacific
Northwest.
Donna Calhoun
Title: Solving the Serre-Green-Naghdi equations for modeling shallow,
dispersive geophysical flows
Abstract: The depth-averaged shallow water wave equations
are commonly used to model flows arising from natural hazards. The GeoClaw
code, developed by D. George, R. J. LeVeque, M. J. Berger, K. Mandli and
others is one example of a depth-averaged flow solver now widely used for
modeling tsunamis, overland flooding, debris flows, storm surges and so on.
Generally, depth averaged flow models show excellent large scale agreement
with observations and can thus be reliably used to predict whether tsunamis
will reach distant coast lines, and if, so can give vital information about
arrival times. However, for other types of flows, dispersive effects missing
from the SWE model play an important role in determining localized effects
such as whether waves will overtop seawalls, or whether a landslide entering a
lake will trigger tsunami-like behavior on the opposite shore. Because of the
importance of these dispersive effects, several depth averaged codes include
dispersive corrections to the SWE. One set of equations commonly used to
model these dispersive effects are the Serre-Green-Naghdi (SGN) equations.
We will present our work to include dispersive correction terms into the
GeoClaw extension of ForestClaw, a parallel adaptive library for Cartesian
grid methods. We will describe the SGN equations and provide an overview of
their derivation, and then show preliminary results on uniform 1d Cartesian
meshes. Comparisons with the SGN solver in Basilisk (S. Popinet) and BoussClaw
(J. Kim et al) will also be shown to verify our model.
Mike Turzewski
Title: Numerical simulations of the Yigong River outburst flood
using GeoClaw software, eastern Himalaya
Authors: Michael D. Turzewski, Katharine W. Huntington, and Randall J. LeVeque
Abstract: High-magnitude outburst floods (>10^5 m^3/s) have
shaped the rugged eastern Himalayan landscape and are a hazard to people and
infrastructure. However, hydraulics during these floods are uncharacterized,
limiting our understanding of the geomorphic impact of this event and the role
of outburst floods in long-term evolution of this landscape. Here we combine
remote and field observations of the 2000 Yigong River landslide-dam outburst
flood with 2D numerical flood simulations using the software GeoClaw. Adaptive
mesh refinement in GeoClaw enables unprecedented resolution of outburst
flood hydraulics through rugged mountainous topography. Simulated hydraulics
are consistent with observations from the flood and are useful to constrain
both hazard and geomorphic processes during the event. The results show that
the hydraulics of outburst floods through this mountainous region differ
from those expected for non-flood flows, in magnitude and in the spatial
patterns of flow speed, direction and shear stress. Our findings highlight the
potential for different magnitude flows to promote not only different amounts,
but different patterns of bedrock erosion, with implications for the role of
prehistoric megafloods in the topographic evolution of the eastern Himalaya.
Speakers for the Waves Sessions
Natalie Sheils
Title: Revivals and Fractalisation in the Linear Free Space Schrödinger
Equation
Abstract: We consider the one-dimensional linear free space Schrödinger
equation on a bounded interval subject to homogeneous linear boundary
conditions. We prove that, in the case of pseudoperiodic boundary conditions,
the solution of the initial-boundary value problem exhibits the phenomenon
of revival at specific ("rational") times, meaning that it is a linear
combination of a certain number of copies of the initial datum. Equivalently,
the fundamental solution at these times is a finite linear combination of
delta functions. At other ("irrational") times, for suitably rough initial
data, e.g., a step or more general piecewise constant function, the solution
exhibits a continuous but fractal-like profile. Further, we express the
solution for general homogeneous linear boundary conditions in terms of
numerically computable eigenfunctions. Alternative solution formulas are
derived using the Uniform Transform Method (UTM), that can prove useful
in more general situations. We then investigate the effects of general
linear boundary conditions, including Robin, and find novel "dissipative"
revivals in the case of energy decreasing conditions.
Vishal Vasan
Title: Fractional derivatives and boundary-value problems
Abstract: Fractional derivatives are as old as integer-order derivatives. In
this talk, I'll introduce a new perspective on fractional derivatives
highlighting their connection to boundary-value problems for partial
differential equations. This perspective readily affords a general framework to
analyze fractional differential equations (FDEs). I'll present two motivating
problems: one arising from the motion of heavy particles in a viscous fluid
and another in anomalous heat transport and super-diffusive systems. The
upshot of our analysis is a much simpler proof for existence of solutions
to some nonlinear FDEs equations, an analysis of the spectrum of fractional
operators as well as a numerical method to compute solutions in an efficient
and accurate manner.
Jeremy Upsal
Title: Determining stability for solutions of integrable PDEs
Abstract: We consider the dynamical stability of periodic and quasi-periodic
solutions to integrable equations that belong to the AKNS hierarchy. The
spectrum of the differential operator obtained through linearization is
important for determining the stability of solutions. When the stability
spectrum is on the imaginary axis, the solutions are spectrally stable. To
determine this stability spectrum, we use the integrability properties of
the underlying equation. For stationary solutions of equations in the AKNS
hierarchy, we show that the real eigenvalues of an auxiliary problem, called
the Lax spectrum, give rise to imaginary, and hence stable, eigenvalues of
the stability problem. In particular, we find that (1) $\mathbb{R}$ is a
subset of the Lax spectrum when the problem is not self adjoint, and (2)
for self-adjoint or non self-adjoint problems, real Lax spectra gives rise
to imaginary, and hence stable, eigenvalues.
Xin Yang
Title: Numerical inverse scattering for the sine-Gordon equation
Abstract: The sine-Gordon equation is a known integrable PDE and was solved by
Kaup using the Inverse Scattering Transform (IST). In 2012, Trogdon, Olver
and Deconinck implemented the IST for the Korteweg-de Vries equation. The
same idea is applied to the sine-Gordon equation, extended so as to account
for the extra singularity appearing
in the IST for the sine-Gordon equation. Time stepping is not required as
opposed to traditional numerical methods. Our numerical experiments show
that the method is spectrally accurate.
John Carter
Title: Particle paths and transport properties of NLS and its generalizations
Abstract: The nonlinear Schrödinger equation (NLS) is well known as a
universal equation in the study of wave motion. In the context of wave
motion at the free surface of an incompressible fluid, NLS accurately
predicts the evolution of modulated wave trains with low to moderate wave
steepness. In this talk, we reconstruct the velocity potential and surface
displacement from NLS coordinates in order to compute particle trajectories
in physical coordinates. We use these particle trajectories to compute
the mean transport properties of modulated wave trains. Additionally, we
present particle trajectories and mean transport properties for the Dysthe
equation and two dissipative generalizations of NLS.
Naeem Masnadi
Title: Experimental investigation of gravity-capillary solitary waves in
deep water
Abstract: I will discuss our experimental work on the generation and dynamics
of three-dimensional gravity-capillary solitary waves (lumps) in deep water. I
will review some of the earlier work on this topic and report our recent
experiments on the periodic generation of lumps near the minimum phase speed
of linear gravity-capillary waves, the nonlinear interaction of these lumps,
and the evolution and decay of free lumps.
Chris Curtis
Title: Nonlinear Waves over Patches of Vorticity
Abstract: In this talk, we present a method for numerically simulating
freely evolving surface waves over patches of vorticity. This is done
via point-vortex approximations and the use of fast-multipole methods for
updating point-vortex velocities. We then present results which show the
impact of varying types of vortex patches on nonlinear-shallow-water wave
propagation. A key result we find is that the more nonlinear a surface
wave, the more robust it is with respects to the influence of submerged
eddies. In contrast, nearly linear waves can be strongly deformed, possibly
to the point of breaking by underwater vorticity patches.
Camille Zaug
Title: Frequency downshift in the Ocean
Abstract: Frequency downshift occurs when a measure of a wave’s frequency
(typically its spectral peak or spectral mean) decreases monotonically. Carter
and Govan (2016) derived a viscous generalization of the Dysthe equation
that successfully models frequency downshift in wave tank experiments for
certain initial conditions. The classical paper by Snodgrass et al. (1966)
shows evidence that narrow-banded swell traveling across the Pacific Ocean also
display frequency downshift. In this work, we test the viscous Dysthe equation
against the Dysthe equation, nonlinear Schrödinger equation, the dissipative
nonlinear Schrödinger equation, and the dissipative Gramstad-Trulsen equation
to see which generalization best models the ocean data reported in Snodgrass
et al. We do so by comparing the Fourier amplitudes, the change in the
spectral peak and spectral mean, and conserved quantities representing mass
and momentum between the ocean measurements and numerical simulations.
Speakers for the Mathematical Finance Session
Bahman Angoshtari
Title: Optimal Dividend Distribution Under Drawdown and Ratcheting
Constraints on Dividend Rates
Abstract: We consider the optimal dividend problem under a
habit formation constraint that prevents the dividend rate to fall below
a certain proportion of its historical maximum, the so-called drawdown
constraint. This is an extension of the optimal Duesenberry's ratcheting
consumption problem, studied by [Dybvig, Review of Economic Studies (1995),
62, 287–313], in which consumption is assumed to be nondecreasing. We
formulate our problem as a stochastic control problem with the objective
of maximizing the expected discounted utility of the dividend stream
until bankruptcy, in which risk preferences are embodied by power utility.
We write the corresponding Hamilton-Jacobi-Bellman variational inequality
as a nonlinear, free-boundary problem and solve it semi-explicitly via the
Legendre transform. We also derive the optimal dividend rate as a function
of the company's current surplus and its historical running maximum of the
dividend rate. Since the maximum (excess) dividend rate will eventually be
proportional to the running maximum of the surplus, "mountains will have
to move" before we increase the dividend rate beyond its historical maximum.
Patricia Ning
Title: Continuous Time Markov Chain Approximation Technique and
its Applications
Abstract: Asset prices are often assumed to follow a
continuous time Markov process with continuous-state space in financial
mathematics. However, there are only a limited number of models have closed
form formulas for certain types of options pricing. In this talk, I will
talk about the method of continuous time Markov chain approximation, which
provides a generally applicable and efficient pricing and hedging framework,
and is able to satisfy the need to use flexible model specifications in
practice. Applications to several kinds of exotic options under different
types of stochastic process models will be discussed.
Jize Zhang
Title: A Relaxed Optimization Approach for Cardinality-Constrained
Portfolios
Abstract: A cardinality-constrained portfolio caps the number
of stocks to be traded across and within groups or sectors. These limitations
arise from real-world scenarios faced by fund managers, who are constrained
by transaction costs and client preferences as they seek to maximize return
and limit risk. We develop a new approach to solve cardinality-constrained
portfolio optimization problems, extending both Markowitz and conditional
value at risk (CVaR) optimization models with cardinality constraints. We
derive a continuous relaxation method for the NP-hard objective, which
allows for very efficient algorithms with standard convergence guarantees
for nonconvex problems. For smaller cases, where brute force search is
feasible to compute the globally optimal cardinality-constrained portfolio,
the new approach finds the best portfolio for the cardinality-constrained
Markowitz model and a very good local minimum for the cardinality-constrained
CVaR model. For higher dimensions, where brute-force search is prohibitively
expensive, we find feasible portfolios that are nearly as efficient as their
non-cardinality constrained counterparts.
Yang Zhou
Title: Top-Down Valuation Framework for Employee Stock Options
Abstract: We propose a new valuation framework for employee
stock options (ESOs). Our approach accounts for vesting period, multiple
early exercises, and sudden job termination. To compute the ESO costs,
we develop several numerical methods to solve the associated systems of
PDEs. We implement and compare different numerical methods, and examine the
effects of various contractual features.
